I present our recent results on the convergence analysis of suitable finite volume methods for multidimensional Euler equations. We have shown that a sequence of numerical solutions converges weakly to a weak dissipative solution. The analysis requires only the consistency and stability of a numerical method and can be seen as a generalization of the famous Lax-equivalence theorem for nonlinear problems. The weak-strong uniqueness principle implies the strong convergence of numerical solutions to the classical solution as long as it exists.
On the other hand, if the classical solution does not exist we apply the so-called K-convergence and show how to compute effectively the observable quantities of a space-time parametrized measure generated by numerical solutions. Consequently, we derive the strong convergence of the empirical averages of numerical solutions to a weak dissipative solution. If time permits I will illustrate a connection to the concept of statistical convergence.